Metrics with Galilean Conformal Isometry

TL;DR

This paper constructs and analyzes metrics in higher dimensions that realize the finite Galilean Conformal Algebra as their isometry group, revealing mostly Ricci-scalar flat solutions with some exotic signatures.

Contribution

It systematically finds higher-dimensional metrics with GCA isometry by coset construction, including non-trivial solutions with exotic signatures.

Findings

Only Minkowskian metric is non-degenerate in standard holographic setting

Found families of metrics with exotic signatures in 4 and 5 dimensions

Most metrics are Ricci-scalar flat

Abstract

The Galilean Conformal Algebra (GCA) arises in taking the non-relativistic limit of the symmetries of a relativistic Conformal Field Theory in any dimensions. It is known to be infinite-dimensional in all spacetime dimensions. In particular, the 2d GCA emerges out of a scaling limit of linear combinations of two copies of the Virasoro algebra. In this paper, we find metrics in dimensions greater than two which realize the finite 2d GCA (the global part of the infinite algebra) as their isometry by systematically looking at a construction in terms of cosets of this finite algebra. We list all possible sub-algebras consistent with some physical considerations motivated by earlier work in this direction and construct all possible higher dimensional non-degenerate metrics. We briefly study the properties of the metrics obtained. In the standard one higher dimensional "holographic" setting, we find that the only non-degenerate metric is Minkowskian. In four and five dimensions, we find families of non-trivial metrics with a rather exotic signature. A curious feature of these metrics is that all but one of them are Ricci-scalar flat.